Properties of the Θ-graph

Since two algorithms for constructing spanners have been presented, it is tempting to try to settle which is the better. In this section properties of the Θ-graph are discussed and compared to those of the greedy algorithm.

The Θ-graph is a t-spanner

The more cones used in the construction of the Θ-graph, the more edges are added. Furthermore, this results in a graph with a lower theoretical bound on the stretch factor. A formula to compute these upper bounds exists, and some of the bounds are listed in the figure below.

Figure 10: Bounding the stretch factor

κ

t

6
7
8
9
10
11
12
13
14
15

∞
∼7.562
∼4.262
∼3.165
∼2.618
∼2.291
∼2.073
∼1.918
∼1.802
∼1.712

Upper bound on the stretch factor (t) of a θ-graph with κ cones.
For κ ≤ 6 there is no theoretical upper bound.

The Θ-graph is thus a t-spanner with t depending on the number of cones.

The Θ-walk property

For κ ≥ 9 a path between vertices in the Θ-graph can be found by always travelling along the edge in the cone containing the destinaton vertice. This metod, referred to as the Θ-walk, computes a path in O(n) time which is even faster than using in example Dijkstra's algorithm.

The path found by the Θ-walk may not be a shortest path, but its detour factor is bounded above by the stretch factor of the graph. This property is usefull in some applications, as it is mentioned in the applications section.

At the website of Petra Specht, University of Magdeburg, an interactive applet constructing of Θ-graph can be found.

Θ-graph vs the greedy algorithm

During the construction, when a number of cones have been chosen, κ can be considered to to be a constant. Thus both the greedy algorithm (in an advanced implementation) and construction of the Θ-graph run in O(n·log n) time, n being the number of vertices in the graph.

Studies of Θ-graphs show, that these tend to contain more edges and have vertices of higher degree that those created by the greedy algortihm. This may not be surprising, since the Θ-graph have up to κ edges from each vertex. On the other hand the diameter of the Θ-graph tends to be low.

Upon desciding which algorithm is better, it is importent to consider what purpose they are going to serve. Each have advantages and disadvantages.